O.Bourget
Singular Continuous Floquet Operator for Periodic Quantum Systems
(374K, pdf)
ABSTRACT. Consider the Floquet operator of a time independent quantum system, acting on a separable Hilbert space, periodically perturbed by a rank one kick: $e^{-iH_0T}e^{-i\kappa T |\phi\ket\bra\phi|}$ where $T$, $\kappa$ are respectively the period and the coupling constant and $H_0$ is a pure point self-adjoint operator, bounded from below. Under some hypotheses on the vector $\phi$, cyclic for $H_0$ we prove the following:
If the gaps between the eigenvalues $(\lambda_n)$ are such that: $\lambda_{n+1}-\lambda_{n}\geq C n^{-\gamma}$ for some $\gamma \in ]0,1[$ and $C>0$, then the Floquet operator of the perturbed system is purely singular continuous T-a.e.
If $H_0$ is the Hamiltonian of the one-dimensional rotator on $L^2({\mathbb R}/T_0{\mathbb Z})$ and the ratio $2\pi T/T_0^2$ is irrational, then the Floquet operator is purely singular continuous as soon as $\kappa T \neq 0(2\pi)$
We also establish an integral formula for the family $(e^{-iH_0T}e^{-i\kappa T |\phi\ket\bra\phi|})_{T>0, \kappa \in {\mathbb R}}.$